# Computing the Hilbert series of an irreducible component of a complete intersection

There's a nice formula for the Hilbert series of any complete intersection of hypersurfaces $$X_1\cap\cdots\cap X_i\subseteq\mathbb{P}^n$$ in terms of the degrees of $$X_1,\ldots,X_i$$. Is there a way to compute the Hilbert series of an irreducible component of $$X_1\cap\cdots\cap X_i$$ in terms of the Hilbert series of $$X_1\cap\cdots\cap X_i$$?

More precisely, let $$k$$ be a field. Suppose $$f_1,\ldots,f_i\in R:=k[x_0,\ldots,x_n]$$ are homogeneous of degrees $$d_1,\ldots,d_i$$. Also suppose that $$(f_1,\ldots,f_i)$$ is a regular sequence in $$R$$, so that the ideal $$I:=(f_1,\ldots,f_i)$$ defines a complete intersection $$X\subseteq\mathbb{P}^n$$ of dimension $$n-i$$. Since $$I$$ is homogeneous, we can look at its Hilbert series $$HS_I$$. Any minimal prime $$\mathfrak{p}$$ associated to $$I$$ is also homogeneous, so we can take its Hilbert series $$HS_\mathfrak{p}$$.

Question: Is there any tractable relationship between $$HS_I$$ and $$HS_\mathfrak{p}$$?

I'm actually interested in computing (an upper bound on) $$\dim_k\mathfrak{p}_{d}$$ in terms of $$d_1,\ldots,d_i$$ for any $$d\leq\min\{ d_1,\ldots,d_i \}$$, and Hilbert series seem like the best tool to use. One could try fiddling around with Gröbner bases, but I haven't been able to see how to make this work as generally as I would like.

• What if each $f_i$ is a linear polynomial times a generic polynomial of degree $d_i-1$, and $\mathfrak p$ is the ideal generated by these linear polynomials? This will make $\dim_k \mathfrak p_d$ very large, regardless of the $d_i$. – Will Sawin Oct 12 '20 at 15:23
• @WillSawin Thanks, this was a helpful example to think through! I can still get the bound I need in this example, and it's given me some new ideas for the general case. – Stephen McKean Oct 12 '20 at 20:24